Abstract
This thesis proposes dynamical models for natural phenomena. The models are investigated analytically for stability criteria and numerically to determine their evolution over time. The first phenomenon we consider is the population dynamics of predator-prey systems, although in this case the predators are small parasites rather than large carnivores. The systems are modelled using delay differential equations, but many different approaches are used within this framework, mainly focusing on the different ways of modelling man's effect on the parasites, whether this should be done discretely or continuously in time and whether it should occur only at particular points in space or time. Secondly we look at the small scale dynamics of granular media, in the form of aeolian sand ripples. Reaction-diffusion equations are used to model the evolution of these ripples over time. We are particularly interested in creating three dimensional models which incorportate the idea of a shadow zone. A shadow zone is an area of the sand bed which is shielded from the incoming wind by a higher altitude area of the bed. In Chapter 1 we propose stage-structured population models for species the adult members of which are subject to culling, with a view to understanding the culling regimes that are likely to result in eradication of the species. A purely time- dependent model is proposed in which culling occurs at particular discrete times, not necessarily equally spaced. Then a reaction-diffusion model is proposed for a situation in which the adults can diffuse; in this model the culling is continuous in time but occurs only at particular discrete points in space. Such a model might be appropriate for pheromone trapping of insects. For both models conditions are obtained that are sufficient for species eradication. In Chapter 2 we propose various stage-structured population models for blowfly strike with the aim of understanding the population dynamics that result in extinction of the blowflies or co-existence of the species. The models include a purely time dependent model, a distributed delay model and a reaction-diffusion model in which the mature blowflies are allowed to move about. We also provide results of numerical simulations of some of these models which show various aspects of the model, including initial conditions for the asymptotic stability of the co-existence steady state and the existence of a threshold value of trapping, above which extinction of the blowfly species will occur. In Chapter 3 we propose models for blowfly strike that use discrete, rather than continuous diffusion in order to simulate the blowflies travelling between a number of farms that contain independent populations of sheep. Models are proposed for both a discrete and an infinite number of farms and conditions are found for extinction of the blowflies. Chapters 5 and 6 of this thesis focus on developing three-dimensional models of aeolian sand ripples which incorporate fully three-dimensional shadowing of the sand bed. Linear stability analyses of these models analytically determine the preferred wavenumbers and growth rates of the ripples. Numerical simulations in Chapter 7 show the evolution of ripples from a perturbed flat bed and illustrate the differences due to the presence of our shadow zone. The effect of gusting during ripple evolution is also investigated showing the instability of the ripple pattern under large changes in the angle of impact and wind direction. It is also shown that a gradual change in wind direction over time will break up an ordered ripple pattern and eventually lead to a new pattern with the crests perpendicular to the final wind direction. Continuing changes in wind direction will lead to disorder or lower amplitude ripples depending on the frequency and magnitude of the gusts.